Secession is akin to divorce; it indicates that any existing constitutional frameworks are deemed insufficient - - - at least by one party - - - to solve the discord. When secession-talk surfaces, it might be blustery, sardonic, or serious.

On Long Island, there is talk of seceding from the State of New York.

The complaint is the not unusual one regarding allocation of resources and taxation. As an editorial in Long Island's Newsday described it:

people are rebellious. But that anger is far more likely to increase musket sales at Civil Wars R Us than it is to create a new State of Long Island. At the Suffolk County Legislature yesterday . . . It became clear the talk had gone a tad too far when a legislator paraphrased Thomas Jefferson's comment that "the tree of liberty must be refreshed from time to time with the blood of patriots and tyrants." Jefferson went on to say that blood is liberty's natural manure. The Suffolk lawmaker left out the manure part, but that's a good word for all this secession rhetoric.

Meanwhile, the Conch Republic just celebrated the twenty-seventh anniversary of its secession, prompted by "a United States Border Patrol Blockade setup on highway U.S.1 at Florida City just to the north of the Florida Keys," isolating the Keys from Florida and the US mainland since US 1 is the only connecting road.
The blockade was a serious issue that entailed litigation, but the Conch Republic has also become a cultural marker of identity.

In Texas, the talk is current and seemingly serious.

On the Texas Secede! website, the answer to the question of whether the Texas state constitution reserves the right to secede is:

No such provision is found in the current Texas Constitution (adopted in 1876) or the terms of annexation. However, it does state (in Article 1, Section 1) that "Texas is a free and independent State, subject only to the Constitution of the United States..."not state (note that it does "...subject to the President of the United States..." or "...subject to the Congress of the United States..." or "...subject to the collective will of one or more of the other States...")

Neither the Texas Constitution, nor the Constitution of the united States, explicitly or implicitly disallows the secession of Texas (or any other "free and independent State") from the United States. Joining the "Union" was ever and always voluntary, rendering voluntary withdrawal an equally lawful and viable option (regardless of what any self-appointed academic, media, or government "experts"—including Abraham Lincoln himself—may have ever said).

But Vermont?

In One of These Things is Not Like the Others?: A Comparative Analysis of Secessionist Movements in Vermont, Quebec, Hawai'i and Kosovo, just posted on ssrn here, Brian M. Lusignan focuses on the Vermont secession movement. Importantly, he sets the stage for his serious consideration by discussing this issue of "experts" that the Texas Secede! website also identifies. Lusignan argues that "despite a widespread belief that support for secession is limited to society’s radical fringe, modern secessionist movements remain surprisingly legitimate." He notes that

a
recent poll revealed that one in five Americans" believe “any state or region has the right to peaceably secede and become an independent republic.” In the South and the East, a quarter of poll respondents believe states have a right to secede. Furthermore, 18% of respondents said they would support a secessionist movement in their own state. Reactions to this poll range from the skeptical to the scathing. Professor Ann Althouse wrote on her blog that “all these people have the law wrong and don't seem to know the basics of the history of the Civil War” and called the results “[f]ascinating(ly stupid).” George Mason Law Professor Ilya Somin responded that “superior military might doesn't prove superior constitutional right” and that a belief in “a right of secession” does not “by itself demonstrates ignorance about either law or American history.”

Lusignan argues that there is

a
deep divide in scholarly opinions on secession. According to Pulitzer Prize-winner Garry Wills, “[s]ecessionist efforts now resemble those of a crackpot group in Texas.” And yet Thomas Naylor, an economics professor at the University of Vermont, founded the secessionist movement the Second Vermont Republic. Naylor’s secessionist book The Vermont Manifesto prompted one legal scholar to suggest that confusion over the constitutionality of secession is caused by “intelligent laymen like Professor Naylor, who have not undergone our [legal] professional socialization” interpreting law and history “in unusual ways.” But Professor Somin is not the only legal scholar who sees the issue of secession as ambiguous. John Remington Graham, a former law school professor, traced the history of secession from England’s Glorious Revolution through the American Revolution and Civil War to the secessionist movement of Quebec. . . . some legal scholars believe that legitimate secession is an open question.

What makes a "secession" legitimate or not is certainly an interesting question, yet whether legitimacy is connected to "constitutionality" is far less clear.